Next: SEABEAM: Filling the empty
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There are at least three ways to fill empty bins.
Two require a roughening operator while
the third requires a smoothing operator which
(for comparison purposes) we denote .The three methods are generally equivalent
though they differ in important details.
The original way in
Chapter is to
restore missing data
by ensuring that the restored data,
after specified filtering,
has minimum energy, say
.Introduce the selection mask operator ,
a diagonal matrix with
ones on the known data and zeros elsewhere
(on the missing data).
Thus
or
| |
(27) |
where we define to be the data
with missing values set to zero by
.
A second way to find missing data is with the set of goals
| |
(28) |
and take the limit as the scalar .At that limit, we should have the same result
as equation (27).
There is an important philosophical difference between
the first method and the second.
The first method strictly honors the known data.
The second method acknowledges that when data misfits
the regularization theory, it might be the fault of the data
so the data need not be strictly honored.
Just what balance is proper falls to the numerical choice of ,a nontrivial topic.
A third way to find missing data is to precondition
equation (28),
namely, try the substitution
.
| |
(29) |
There is no simple way of knowing beforehand
what is the best value of .Practitioners like to see solutions for various values of .Of course that can cost a lot of computational effort.
Practical exploratory data analysis is more pragmatic.
Without a simple clear theoretical basis,
analysts generally begin from and abandon the fitting goal .Implicitly, they take .Then they examine the solution as a function of iteration,
imagining that the solution at larger iterations
corresponds to smaller .There is an eigenvector analysis
indicating some kind of basis for this approach,
but I believe there is no firm guidance.
Before we look at coding details for the three methods
of filling the empty bins,
we'll compare results of trying all three methods.
For the roughening operator ,we'll take the helix derivative .This is logically equivalent to roughening with the gradient because the (negative) laplacian operator is
.